in this image, we compute g(w* a +b ) and the output is scaler value not vector

so how does layer compute g(w* a +b ) → 1/1+e power(-z)

and convert (a) vector to scaler value (output of neural network)

In general, w and a are vectors, so the multiplication operator is a dot product. The result of the dot product of two vectors is a scalar (by definition).

This result is then passed through the function g(), which traditionally means sigmoid.